For the Hydrogen atom, higher orbital numbers are generally higher energy, but if the electron is far enough away, the expansion means the electron will have a little less energy than in a static universe. With a high enough orbital number, or for a free electron far away, that effect will win out, making it energetically favorable for an electron to move to that level or escape. Given that it’s a 1 part in 10[sup]136[/sup] effect, it will take a very[sup]136[/sup] long time, but eventually the hydrogen atom will tunnel out, and decay into an electron and a proton far apart. (Ignoring the half life of the proton itself.)
I can’t quite dissect the reasoning here to speak to it directly, but the conclusion is off. Transitioning to a higher energy state (either bound or unbound) requires energy, and expansion doesn’t provide that. Even accelerating expansion just changes the nature of the bound states; it doesn’t add energy to the system.
You can think of it as another force in the equation. The particles are attracted by the Coulomb force. They are also attracted (ever so slightly) by usual gravity. And they are repelled by “unusual” gravity. This third term just joins the first two, and the result is some bound system that is just slightly less than it would be without the third term.
Only if the universe is expanding and that expansion is accelerating and that acceleration is increasing with time will the hydrogen atom be pulled apart. However, in that scenario (the “phantom energy” scenario) the ripping isn’t analogous with tunneling.
Won’t the additional term from the expansion seen by the electron look something like -ar^2 with a barely greater than zero, giving for the Hydrogen atom a potential of -1/r - a*r^2? That additional term dominates for large enough r.
It wouldn’t necessarily be as simple as -ar^2, but it would be concave downward, and unbounded below.
Hmm… I don’t think so for mere expansion, but for w=-1 (accelerating but not hyper-accelerating) I think you’re right. If I have a chance this weekend (unless you or someone else gets to it first), I’ll get an order-of-magnitude estimate for how far out the inflection point in the potential would be assuming w=-1. For instance, would it even be within the observable universe? I just scanned briefly for any useful discussion in the literature, but I didn’t see anything of note.
(A full treatment of this phenomenon would, of course, need a working theory of quantum gravity, as it’s manifestly relativistic and quantum mechanical. But the semi-classical/semi-Newtonian thought experiment is interesting.)
Approximating the cosmological constant as a Newtonian potential, by my hasty calculation the repulsive force of a cosmological constant that agrees with current estimates only becomes greater than the gravitational attraction of a proton at distances greater than 33 cm and greater than the electrostatic force exerted by a proton on an electron at distances greater than 0.1 light-years.
Actually I realized I missed a constant out so the distance where the force due to the cosmological constant becomes greater than the electrostatic force exerted on an electron by a proton is about 800,000,000 m which is roughly a quarter of the distance to the Moon and a lot less than 0.1 light-years
Bump.
Dumb questions: c. ~1990 we didn’t need a cosmological constant and the term “Dark Energy” had yet to be coined. But it seems to me that the OP’s question still applied. So is it proper to pitch this issue as one of opposing forces?
Also, set aside intra-atomic effects. As space in the universe expands, do objects such as cars, bricks and bowling balls grow as well? Or does the preceding apply at the molecular, etc. level as well?
I wrote about this in post #18. The first 3 paragraphs cover the situation without dark energy/cosmological constant. Including dark energy or a cosmological constant doesn’t necessarily invalidate what I wrote, it just adds an extra factor.
Viewing dark energy and gravity as opposing forces is only an approximation, obviously with no dark energy there is no term to approximate as a repulsive force.
The theoretical effect of expansion on objects on the scale of galaxies is still an open question as for the reasons I stated in post #18 and the fact that not a huge amount on the nature of the dark energy is known.
As the OP, I think a lot of the questions have been answered. I was being deliberately vague in the question and skirting about the question of constant versus accelerating expansion, which as later answers showed is actually a critical differentiation. Accelerating expansion adds energy, constant does not.
But the crucial thing is the assumption of scale and homogeneity of expansion. I was thinking in terms of a hybrid model where space is sort of pure relativistic, and the quantum effects live on it. But you can’t do this. Spacetime is a theory of gravity, and imposing quantum mechanics on this implies that we have a theory of quantum gravity - which we don’t. So the thought experiment reduces to a situation we know is wrong, but might be interesting.
What I suspect is also hinted at is that we don’t know how mass/energy affects the local expansion of space. A simple model might say that it simply opposes it as a gravitational force, but we don’t know, and it may be that it wipes out local expansion totally, and we only see expansion of space occurring where space is very very very flat - in intergalactic cluster gaps perhaps. One wonders if there are clues possible from some of the very deep galactic surveys for the Hubble constant, or from the cosmic background surveys that might correlate enough.
No doubt there are some really interesting and deep question here.
In fact without a dark energy-like term expansion must be decelerating: either unboundedly or approaching some constant.
Your first intuition was right: there is absolutely nothing stopping you from doing quantum physics in a background of curved spacetime. This is the regime that Stephen Hawking used to derive Hawking radiation for example. The problem applying it to this situation is that the curvature in the small patch of spacetime that describes an atom is so small that it is essentially flat.